Cartan subgroup
In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group [math]\displaystyle{ G }[/math] over a (not necessarily algebraically closed) field [math]\displaystyle{ k }[/math] is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If [math]\displaystyle{ k }[/math] is algebraically closed, they are all conjugate to each other. [1]
Notice that in the context of algebraic groups a torus is an algebraic group [math]\displaystyle{ T }[/math] such that the base extension [math]\displaystyle{ T_{(\bar{k})} }[/math] (where [math]\displaystyle{ \bar{k} }[/math] is the algebraic closure of [math]\displaystyle{ k }[/math]) is isomorphic to the product of a finite number of copies of the [math]\displaystyle{ \mathbf{G}_m=\mathbf{GL}_1 }[/math]. Maximal such subgroups have in the theory of algebraic groups a role that is similar to that of maximal tori in the theory of Lie groups.
If [math]\displaystyle{ G }[/math] is reductive (in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser [2] and thus Cartan subgroups of [math]\displaystyle{ G }[/math] are precisely the maximal tori.
Example
The general linear groups [math]\displaystyle{ \mathbf{GL}_n }[/math] are reductive. The diagonal subgroup is clearly a torus (indeed a split torus, since it is product of n copies of [math]\displaystyle{ \mathbf{G}_m }[/math] already before any base extension), and it can be shown to be maximal. Since [math]\displaystyle{ \mathbf{GL}_n }[/math] is reductive, the diagonal subgroup is a Cartan subgroup.
See also
References
- ↑ Milne (2017), Proposition 17.44.
- ↑ Milne (2017), Corollary 17.84.
- Borel, Armand (1991-12-31). Linear algebraic groups. ISBN 3-540-97370-2.
- Lang, Serge (2002). Algebra. Springer. ISBN 978-0-387-95385-4.
- Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press, 2017, doi:10.1017/9781316711736, ISBN 978-1107167483
- Hazewinkel, Michiel, ed. (2001), "Cartan subgroup", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
- Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics, 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7
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